A car comes to a bridge during a storm and finds the bridge washed out. The driver must get to the other side, so he decides to try leaping it with his car. The side the car is on is 21.5 m above the river, whereas the opposite side is a mere 1.5 m above the river. The river itself is a raging torrent 69.0 m wide. Required:
a. How fast should the car be traveling just as it leaves the cliff in order just to clear the river and land safely on the opposite side?
b. What is the speed of the car just before it lands safely on the other side?

Comedians like to joke that the reason we haven’t been visited by intelligent life from elsewhere in the universe is that aliens have been monitoring earth’s broadcasts of intellectually embarrassing tv programs, like gilligan’s island, fear factor, the jersey shore, and the jerry springer showand so consider us far too primitive to merit a visit. let’s check the assertion that aliens could have been receiving them. tv programs are broadcast at a frequency of about 100mhz with about 100kw of total power in 30frames per second, which emanatesroughly uniformly in all directions. assume that interstellar space transmits these broadcasts without attenuation. no matter how smart they are, aliens would require at least one photon per frame to interpret our signals. findthe number of photons per unit time per unit area reaching a receiver on a possible planet in the nearest star system, which is about 4 light-years away (a light-year is the distance light travels in a year). if aliens aimed a receiver or detector directly at earth, how big (in diameter) would it have to be to receive a photon per frame

Multiply or divide to make this english distance conversion. 174 inches = feet (14.5, 58, 2,088, 23)

Suppose that f : rn → rm and that a ∈ k, where k is a connected subset of rn . suppose further that for each x ∈ k there exists a δx > 0 such that f(x) = f(y) for all y ∈ bδx (x). prove that f is constant on k; that is, f(x) = f(a) for all x ∈ k